Temperature dependence of the Gibbs state in the random energy model

We consider the problem of temperature dependence of the Gibbs states in two spin-glass models: Derrida' s Random Energy Model and its analogue, where the random variables in the Hamiltonian are replaced by independent standard Brownian motions. For both of them we compute in the thermodynamic limit the overlap distribution; Sigma(i=1)(N) sigma(i)sigma'(i) / N is an element of [-1, 1] of two spin configurations sigma, sigma' under the product of two Gibbs measures, which are taken at temperatures T, T' respectively. If T not equal T' are fixed, then at low temperature phase the results are different for these models: for the first one this distribution is D(0)delta(0) + D(1)delta(1), with random weights D-0, D-1, while for the second one it is delta(0). We compute consequently the overlap distribution for the second model whenever T-T --> 0 at different speeds as N --> infinity.